Lol another analog problem psp owner. Let me tell you my story with analogs:
I once accidentally dropped my phat psp. It slipped out of my pocket and fell hard on to the floor.
Four things happened, my memory stick flew out, wifi switch broke, the psp temporarily couldn't be started and the analog f*cked up.

Eventually the psp could start, and i managed to replace the wifi switch with a new one, despite being slightly looser than usual. Luckily my memory stick was still usable. The only thing i couldn't repair was the damned analog stick.
From time to time, it will keep moving to one corner of the screen by itself.
I asked many forums but they couldn't help much. So i dismantled the thing, and checked it out for myself. Seemed nothing wrong. I dusted some parts, put them back together and voila! temporary fix found. it lasted for a month or so, then the problem starts coming again.
So now every month, i take take apart the psp, dust it, and push the transparent rubber thingy HARD onto the motherboard and its temporarily fixed.
The temporary fix only lasts that long because im very calm when playing games. If given someone who is very kan cheong when playing and rough in handling the machine, the fix doesnt last long.

For your problem however, are you sure you didnt touch or damage the motherboard or the small transparent conductive rubber thing stuck to the motherboard?
The conductive rubber is VERY sensitive. It reacts even when theres dust on it. Also, make sure you screw the analog on properly, not too loose, not too tight either. any of that will cause the analog to malfunction. In your case, the gold plates on the analog probably did not touch the conductive rubber, loosen it a bit and try again.
G'Luck cracking your head ^^